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?? 2 ? d?
This reasoning can be summarized in the following:
Result 4. (i) The function

?(x) = u(x) exp(ig(u(x)))

is a solution of (1) whenever g is a solution of (24) and w(x) = f (u(x)) is a solution
of
?
2w = , wµ wµ = ?
w
with f given by (23).
(ii) The function

?(x) = u(x) exp(ig(u(x)))

is a solution of (1) whenever g is a solution of (26) and w(x) = f (u(x)) is a solution
of
?N
2w = , wµ wµ = ?
w
with f given by (25) for N = 1.
We treat equations (24) and (26) relating g to the nonlinearity F as before: we
assume a form for g and treat the equations as determining F . Taking g(u) = u? ,
Solutions of nonlinear d’Alembert equations 333

we have the following examples of the wave equation, exact solution and relation
between u and w:
N = 1, ? = ?2.
??
?? 2 ? + 1 ?2?
2? + 2 |?|?2 1 + |?| |?|?+2 ? = 0,
exp
2
C ? ?(? + 2)
?(x) = u(x) exp[iu(x)? ],
1/(?+2)
?(? + 2) w
u= ln ,
? C
where w is a solution (listed in table 1) of the compatible system
?
2w = , wµ wµ = ?.
w
N = 1, ? = ?2.
2N/(N ?1)
(N ? 1)?
? + 1 ?2?
?2
2? + ?? |?| |?| C? |?|?+2
2
1+ ? = 0,
?2 ?(? + 2)
?(x) = u(x) exp[iu(x)? ],
1/(?+2)
?(? + 2)
(C ? w1?N )
u= ,
(N ? 1)?
where w is a solution (listed in table 1) of the compatible system
?N
2w = , wµ wµ = ?.
w
If we choose ? = ?1, N = 2, C = 0, then we find that the wave equation is
2? + ?? ?2 |?|2 ? = 0 (26)
with the exact solution
i
?(x) = u(x) exp
u(x)
and
u(x) = ?w(x),
where w solves
2?
2w = , wµ wµ = ?.
w
Equation (27) is of some interest: of all the possible nonlinearities F (|?|), the
nonlinearity F (|?|) = |?|2 gives the widest possible symmetry group, admitting
the conformal group. Equation (27) (and indeed equation (1)) can be reduced to the
nonlinear Schr?dinger equation in (1 + 2)-dimensional time-space (see [1]) with the
o
same nonlinearity. This equation also admits the widest possible symmetry group
for nonlinearities of the given type. It can be reduced to the (1 + 1)-dimensional
nonlinear Schr?dinger equation with the same nonlinearity, and this equation has
o
334 P. Basarab-Horwath, N. Euler, M. Euler, W.I. Fushchych

soliton solutions (the well known Zakharov–Shabat soliton). Using this soliton, we
can construct a new type of solution of the hyperbolic wave equation (27). Of course,
this does not imply that (27) has soliton solutions located in three-dimensional space.
3. Conclusion
We have demonstrated an approach which can give new exact solutions of some
nonlinear wave equations of the same type as (1). The novelty in our approach lies
in the fact that we exploit the compatibility conditions for the d’Alembert–Hamilton
system to dictate the type of nonlinearity and the exact solution(s). Moreover, some
of the equations we obtain appear to be of interest in physics, but we are unable to
make any statement about the physical nature of the exact solutions we obtain, as
our approach has not used any physical criteria to single out any type of solution.
Of course, this is not the only approach possible; we could, for instance, reduce (1)
to the Schr?dinger equation (as in [1]) and then apply a similar method to this
o
new equation in the amplitude-phase representation. Also, it is possible to consider
a more general connection between the amplitude and phase, such as u = G(vµ vµ )
for some function G. This leads to a system involving the Born–Infeld equation,
which has a very wide symmetry group, and we obtain new exact solutions of (1).
This differential connection between amplitude and phase will of course be important
when we allow nonlinearities dependent on derivatives, such as F (|?|, ?? ?µ ). We
µ
will report on this work in a forthcoming paper.
Acknowledgments
WIF thanks the Soros Foundation and the Swedish Natural Science Research
Council (NFR grant R-RA 09423-314) for financial support, and the Mathematics
Department of Link?ping University for its hospitality. PB-H thanks the Wallenberg
o
Fund of Link?ping University and the Tornby Fund for travel grants, and the Mathe-
o
matics Institute of the Ukrainian National Academy of Sciences in Kiev for its hospi-
tality.

1. Basarab-Horwath P., Barannyk L., Fushchych W., J. Phys. A: Math. Gen., 1995, 28, 1–14.
2. Collins C.B., Math. Proc. Camb. Phil. Soc., 1976, 80, 165–187.
3. Collins C.B., Math. Proc. Camb. Phil. Soc., 1976, 80, 349–355.
4. Collins C.B., J. Math. Phys., 1980, 21, 349–355.
5. Collins C.B., J. Math. Phys., 1983, 24, 22–28.
6. Fushchych W.I , Serov N.I., J. Phys. A: Math. Gen., 1983, 16, 3645–3658.
7. Fushchych W.I., Shtelen W., Serov M., Symmetry analysis and exact solutions of equations of
nonlinear mathematical physics, Dordrecht, Kluwer, 1993.
8. Fushchych W.I., Yehorchenko I.A., J. Phys. A: Math. Gen., 1989, 22, 2643–2652.
9. Fushchych W.I., Zhdanov R.Z., On some exact solutions of nonlinear d’Alembert and Hamilton
equations, Preprint Institute for Mathematics and Applications, University of Minnesota, 1988.
10. Fushchych W.I., Zhdanov R.Z., Yehorchenko I.A., Mathematical Analysis and Applications, 1991,
161, 352–360.
11. Grundland M.A, Harnad J., Winternitz P., J. Math. Phys., 1984, 25, 791–805.
12. Grundland M.A., Tuszynski J.A., J. Phys. A: Math. Gen., 1987, 20, 6243–6258.
13. Grundland M.A., Tuszynski J.A., Winternitz P., Phys. Lett. A, 1987, 119, 340–344.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 335–337.

Planck’s constant is not constant
in different quantum phenomena
O. BEDRIJ, W.I. FUSHCHYCH
i , h ii i ii
i ii. i i ,
i , ii .

At present, it is a generally accepted axiom that the Planck’s constant

h = 6.626 · 10?34 J · s (1)

has the same meaning and value in electrodynamics, mesodynamics, quantum theory,
theory of quarks, gravidynamics, etc. Planck’s fundamental quantum hypothesis, put
forward for the explanation of the energy spectrum of black body radiation, is ad hoc
employed in all quantum physics.
In [1] we have suggested the following hypothesis: the fundamental value of
Planck’s constant h in mesodynamics is considerably different from (1). This assumpti-
on, for example, can be explained by the fact that in mesodynamics, not a photon but a
meson is emitted, which mass does not equal zero. There are no fundamental grounds
to assume that h in mesodynamics has to have the value of (1) [1–4].
In this short note we focus on the equation of motion for the fundamental particles
(e — electron, p — proton, n — neutron) based on the aforementioned hypothesis.
Schr?dinger equations for electron, proton and neutron have the following form in our
o
approach:
2
??e
= ? e ??e + Vc (x)?e , (2)
ie
?t 2me
2
??p p
=? (3)
ip ??p + Vp ?p ,
?t 2mp
2
??n
= ? n ??n + Vn ?n , (4)
i n
?t 2mn

?2 ?2 ?2
?? + + ,
?x2 ?x2 ?x2
1 2 3

where

=, =, =,
e p n

— Planck’s constant for electron (electrodynamics); p — Planck’s constant for
e
proton (mesodynamics); n — Planck’s constant for neutron (mesodynamics).
ii , 1995, 12, . 26–27.
336 O. Bedrij, W.I. Fushchych

The Ve , Vp , Vn potentials are assumed to be depended on he , hp and hn , the wave
functions ?e , ?p and ?n as well as the coordinates of particles.
In addition, let us consider Poincar? invariant equations of motion for meson and
e
for e, p, n. As is known, the energy of an elementary particle is defined by formulae:
E 2 = c2 p2 + m2 c4 , p2 = p2 + p2 + p2 , (5)
a a 1 2 3

where m is the mass of a particle; c is the velocity of light in vacuum; pa is
momentum. Formulae (5) give us the following Poincar?-invariant equations for
e
particles
2
2? u
? =? 22
+ m2 c4 + Vµ u, (6)
µc ?
µ µ
?t2
??e ?
?i e ?0 ?k + me c2 ?0 ?e + Ve ?e , (7)
i =
e
?t ?xk
??p ?
?i + mp c2 ?0 ?p + Vp ?p , (8)
i = p ?0 ?k
p
?t ?xk
??n ?
?i + mn c2 ?0 ?n + Vn ?n , (9)
i = n ?0 ?k
n
?t ?xk
where ?e , ?p , ?n are four-component wave functions; u is a scalar wave function
for meson with mass mµ ; ?µ are 4 ? 4 Dirac’s matrices. Equations (7), (8) and
(9) are Dirac’s equations with different Planck’s constants, Ve = Ve (?e , ?p , ?n , x, t),
Vp = Vp (?e , ?p , ?n , x, t), Vn = Vn (?e , ?p , ?n , x, t).
Consequently, to describe interactions between electron and proton, electron and
neutron, etc., it is necessary to use different values for e , p and n in equations
(6)–(9).
A phenomenological approach, proposed in [2–4] for determining fundamental
constants and based on a few known constants, gives us the following values [2, 3]:
he = 6.626 · 10?34 J · s, hp = 2.612 · 10?30 J · s, hn = 2.668 · 10?30 J · s.
Obviously, because he , hp and hn enter most of quantum relationships, we must re-
view the standard theoretical schemes and possibly explore new physical experiments.
This fundamental challenge will take time. Our main objective is to show a new
possibility for description of interactions of particles which is related to a new value
of h.
According to [5, 6], formulae (5) can be used for nonlinear generalization of
equations of motion for elementary particles. Assume that in formulae (5) c is not
constant but a function of field (or a functional with respect to fields)
? ?
? ? ?? ? ?? . (10)
c = c ??,
?xµ ?xµ
Therefore, we can obtain from (10) a nonlinear equation of the type (6)–(8). This
assumption means that velocity of a signal is a function of field [5, 6) and not
a constant, as is presently accepted for the velocity of light in vacuum. The latter
statement is a cornerstone of modern quantum physics. We should like to emphasize
that here we have discussed a new glance on this fundamental point.
A more detailed development of these ideas will be published elsewhere.
Planck’s constant is not constant in different quantum phenomena 337

1. Bedrij O., Fushchych W.I., Fundamental constants of nucleonmeson dynamics, . AH pa,
1993, 5, 62–65.
2. Bedrij O., Fundamental constants in quantum electrodynamics, . AH pa, 1993, 3,
40–45.
3. Bedrij O., Connection of ? with the fine structure constants, . AH pa, 1994, 10, 63–66.
4. Bedrij O., Scale invariance, unifying principle, order and sequence of physical quantities and
fundamental constants, . AH pa, 1993, 4, 67–74.
5. Fushchych W., New nonlinear equations for electromagnetic field having a velocity different from c,
. AH pa, 1992, 4, 24—27.
6. Fushchych W., Nikitin A., Symmetries of equations of quantum mechanics, New York, Allerton
Press, 1994, 458 p.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 338–356.

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